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We introduce two classes of Metropolis-Hastings algorithms for samplingtarget measures that are absolutely continuous with respect to non-Gaussianprior measures on infinite-dimensional Hilbert spaces. In particular, we focuson certain classes of prior measures for which prior-reversible proposalkernels of the autoregressive type can be designed. We then use these proposalkernels to design algorithms that satisfy detailed balance with respect to thetarget measures. Afterwards, we introduce a new class of prior measures, calledthe Bessel-K priors, as a generalization of the gamma distribution to measuresin infinite dimensions. The Bessel-K priors interpolate between well-knownpriors such as the gamma distribution and Besov priors and can model sparse orcompressible parameters. We present concrete instances of our algorithms forthe Bessel-K priors in the context of numerical examples in density estimation,finite-dimensional denoising and deconvolution on the circle.

Two Metropolis-Hastings algorithms for posterior measures with non-Gaussian priors in infinite dimensions